research article quantitative diagnosis of rotor vibration...

Research ArticleQuantitative Diagnosis of Rotor Vibration Fault Using ProcessPower Spectrum Entropy and Support Vector Machine Method

Cheng-Wei Fei1 Guang-Chen Bai1 Wen-Zhong Tang2 and Shuang Ma3

1 School of Energy and Power Engineering Beijing University of Aeronautics and Astronautics Beijing 100191 China2 School of Computer Science and Engineering Beijing University of Aeronautics and Astronautics Beijing 100191 China3 School of Life Science Beijing Normal University Beijing 100875 China

Correspondence should be addressed to Cheng-Wei Fei feicw544163com

Received 19 November 2013 Revised 8 March 2014 Accepted 10 March 2014 Published 31 March 2014

Academic Editor Valder Steffen

Copyright copy 2014 Cheng-Wei Fei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To improve the diagnosis capacity of rotor vibration fault in stochastic process an effective fault diagnosis method (named ProcessPower Spectrum Entropy (PPSE) and Support Vector Machine (SVM) (PPSE-SVM for short) method) was proposed The faultdiagnosis model of PPSE-SVM was established by fusing PPSE method and SVM theory Based on the simulation experimentof rotor vibration fault process data for four typical vibration faults (rotor imbalance shaft misalignment rotor-stator rubbingand pedestal looseness) were collected under multipoint (multiple channels) and multispeed By using PPSE method the PPSEvalues of these data were extracted as fault feature vectors to establish the SVMmodel of rotor vibration fault diagnosis From rotorvibration fault diagnosis the results demonstrate that the proposed method possesses high precision good learning ability goodgeneralization ability and strong fault-tolerant ability (robustness) in four aspects of distinguishing fault types fault severity faultlocation and noise immunity of rotor stochastic vibration This paper presents a novel method (PPSE-SVM) for rotor vibrationfault diagnosis and real-time vibration monitoring The presented effort is promising to improve the fault diagnosis precision ofrotating machinery like gas turbine

1 Introduction

Vibration is a momentous fault source of rotating machinerylike an aeroengine and seriously impacts on the securityand reliability of machine system operation [1] With thedevelopment of the high performance and high reliability ofrotating machinery vibration fault needs to be predicted andinhibited early [1ndash3] Therefore how to predict and controlrotor vibration faults is one of hot issues in preventing thefailures of mechanical system which leads to the advance offeasible and effective fault diagnosismethods [3ndash8] Howevermost of present vibration analysis techniques need a massof vibration samples to establish a fault diagnosis model todiagnose the vibration faults from a qualitative perspectiveIn fact these fault analysis methods possess some blind-ness in fault diagnosis which seriously influences diagnosisaccuracy due to being short of describing diagnosis resultsfrom a process and quantitative perspective Meanwhile itis always difficult to gain large number of vibration fault

data In order to improve the validity of fault diagnosisthe process and quantitative factors should be consideredto ascertain fault types failure severity fault location andeven development tendencyThedevelopment of informationentropy theory is promising to quantitatively analyze rotorvibration conditions and describe the uncertainty degree ofvibration signal [6 9 10] Information entropy technique asa process fault diagnosis method has been widely researchedand applied to machinery fault diagnosis [8 10ndash12] Speciallyfor quantitative diagnosis of rotor vibration faults processinformation entropy method also called as Process PowerSpectrum Entropy (PPSE) method was proposed and provedto be effective [11] Although these techniques are feasible indiagnosing some simple faults due to the complexity of vibra-tion signals for large rotating machinery information fusiontechnology is required to synthetically deal with multisensorinformation in order to ameliorate fault analysis precision[3 4 13 14] Support Vector Machine (SVM) is a newintelligent pattern recognitionmethod which uses structural

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 957531 9 pageshttpdxdoiorg1011552014957531

2 Shock and Vibration

risk minimization instead of empirical risk minimization tosolve small-sample nonlinear high-dimensional problemsand so on SVMhas been proved to embody various strengthsof complete theory good adaptability global optimizationshort training time and good generalization ability in faultdiagnosis [3 8 14ndash18]

The purpose of this present study attempts to proposean effective approach which is Process Power SpectrumEntropy-SVM (PPSE-SVM) method for rotor vibration faultdiagnosis based on PPSE method and SVM theory Thismethod is promising to resolve the randomization of vibra-tion fault and the difficulty of extracting vibration faultsamples Based on rotor vibration simulation workbenchfour typical faults are simulated and their vibration dataare gained The PPSE values (ie information features) ofthese data are calculated based on PPSE method to constructeigenvectors as fault diagnosis samples SVM fault diagnosismodel was established based on the fault diagnosis samplesThe feasibility and validity of PPSE-SVMmethod are verifiedby diagnosing rotor vibration fault types failure severity faultpoint and noise immunity

2 Process Power Spectrum Entropy Method

Information entropy was first propounded by Shannon toevaluate information capacity [3 6ndash12] Assuming thatM is aLebesgue space with an algebra 120575 generated by a measurableset 119867 and a measure 120583 (120583(M) = 1) and the space Mmay be decomposed as a limited partitioning A = (119860

119894)

which is an incompatible set satisfying M = ⋃119899

119894=1119860119894and

119860119894⋂119860119895= 0 forall119894 = 119895 based on information entropy theory the

information entropy E(A) of A is denoted as [3]

119864 (A) = minus119899

sum

119894=1

120583 (119860119894) log120583 (119860

119894) (1)

where 120583(119860119894) is the measurement of sample 119860

119894 119894 = 1 2 119899

As shown in (1) information entropy 119864 shows actuallythe chaotic degree of uncertain factors in a system Moredisorder and randomness of the system always lead to that thecorresponding information entropy values become greaterand vice versa [7]

The Power Spectrum Entropy (PSE) values of rotor vibra-tion signal can be extracted from the features of vibrationsignals in frequency domain from an energy perspectivewhen 119883(120596) is the discrete Fourier transform of a single-channel signal 119909

119905 as follows

119883 (120596) =1

2119899120587

119873minus1

sum

119905=1

119909119905119890minus119895120596119905

(2)

Thus the power spectrum of119883(120596) is defined by

119878 (120596) =1

2120587119873|119883 (120596)|

2 (3)

due to the energy conserve law in the signal transformationprocess from the time domain to the frequency domain [11]that is

sum1199092(119905) Δ119905 = sum |119883 (120596)|

2Δ120596 (4)

Equation (4) reveals that the total energy of signal equalsthe sum of the subenergy of each frequency componentTherefore the power entropy 119878 = 119878

1 1198782 119878

119873 of every

natural frequency is regarded as one original signal partitionthe corresponding information entropy (also called as PPSE)can be defined by

119864119891= minus

119873

sum

119894=1

119902119894log 119902119894 (5)

where the subscript 119891 stands for the frequency domain 119902119894is

the ratio of the 119894th power spectrum to the whole spectrumwhich is denoted by

119902119894=

119878119894

sum119873

119894=1119878119894

(6)

Equation (5) is called the Power Spectrum Entropy (PSE) ofsignal 119878 When PSE is applied to fault diagnosis the methodis called PSE method PSE explains the spectrum structureof single-channel vibration signal The more uniformity thevibration energy distributes in the whole frequency compo-sition the more complex the signal is and the greater theuncertainty degree is When the process of rotor speed-up(or -down) is constituted from multiple rotation speeds thePSE values of different rotation speeds describe the vibrationcondition of whole speed-up (or -down) process Hence PSEmethod is called also Process PSE (PPSE) method

Due to the lack of quantitative indexes for informa-tion entropy information entropy values cannot be directlyapplied to diagnose rotor fault The PPSE presenting theconditions of rotor vibration is promising to be used todistinguish rotor faults However the different momentsand measuring points of vibration waveform always resultin that PPSE values have a certain distribution range foreach fault and there is invariably some overlap amongthe PPSE distribution ranges of different faults [7ndash9] Forexample the PPSE values of four typical faults from multiplesimulation experiments are listed and shown in Table 1Theirdistribution ranges have large overlap regions When thePPSE value of an unknown fault is 50 it is difficult to judge towhich class the unknown fault belongs Therefore a perfectand effective identification method needs to be proposedto address this issue SVM method is an intelligent patterrecognition method and is promising to address this overlapproblem of the PPSE values of different fault modes

3 SVM Method

31 Fundamental Theory Support Vector Machine (SVM)[15ndash17] is a newmachine learningmethodwhich is promisingto address the realistic issues of small sample nonlinear andhigh-dimensional pattern recognition in fault diagnosis andhas the advantages of complete theory good adaptabilityglobal optimization short training time and good general-ization ability SVM adopts small support vectors in trainingsamples standing for the whole vector set to establish a SVMclassifier for fault diagnosis Assuming that the fault trainingsample set is (119909

119894 119910119894) 119894 = 1 2 119871 where 119909 isin 119877

119899 and

Shock and Vibration 3

Table 1 The PPSE distribution range of rotor vibration faults

Fault type PSE distribution rangeRotor imbalance 3659sim5260Shaft misalignment 3945sim5488Rotor rubbing 4357sim5618Pedestal looseness 3578sim5526

119910 isin | minus 1 +1| and 119892(119909) = (120596 sdot 119909) + 119887 is the general formof linear discriminated function in an N-dimensional spacethe optimal hyper plane (120596 sdot 119909) + 119887 = 0 is able to correctlyclassify the data in the sample set The distance betweensupport vector and hyperplane is 1120596 thus the problem ofsearching for hyper plane may be translated into the questionof solving quadratic programming by

min 1

2(120596 sdot 120596) + 119862

119899

sum

119894=1

120577119894

st 119910119894[(120596 sdot 119909

119894) + 119887] + 120577

119894ge 1

(7)

where120596 and 119887 are undetermined coefficients 120577119894is a relaxation

factor and 119862 is a penalty factor standing for the compromiseof the classification interval and error rate

According to the optimization theory the objective func-tion has a unique global optimal solution because objectivefunction and constraint conditions are convex functions Forlinear programming problem with Lagrange algorithm and119910119894[(119908 sdot 119909

119894) + 119887] = 1 the decision function of optimal

hyperplane is

119891 (119909) = sgn [120596 sdot 119909 + 119887lowast] = sgn(sum119894

119886119894119910119894119909119894sdot 119909 + 119887

lowast) (8)

For nonlinear programming problem a kernel functionneeds to be introduced to determine the optimal hyperplaneof feature vectors The corresponding decision function is

119891 (119909) = sgn(sum119894

119886lowast

119894119910119894119896 (119909119894 119909) + 119887

lowast) (9)

The algorithm thought of optimal hyperplane in SVM isthat the input vector x is mapped into a high-dimensionalfeature space Z by preselected nonlinear mapping to con-struct optimal classification hyperplane in high-dimensionalfeature space Z The output results of SVM classificationfunction are a linear combination of intermediate nodes eachcorresponding to one support vector as shown in Figure 1

In Figure 1 kernel functions are promising to avoidcomplex calculation in high-dimensional feature space gen-erally including linear kernel function polynomials kernelfunction RBF kernel function and Sigmoid kernel function[3 16ndash19] This present study selects RBF kernel functiondenoted by

119896 (119909119894 119909) = exp[minus

1003817100381710038171003817119909 minus 11990911989410038171003817100381710038172

1205902] (10)

where 120590 is the kernel width

y

y1a1lowast

y2a2lowast y3a3

lowast ysaslowast

K(x1 x) K(x2 x) K(x3 x) K(xs x)

xt xtminus1 xtminus2 xtminus(mminus1)

middot middot middot


Φ

y = sgn(s

sumi=1

ailowastyiK(xi x) + blowast)

Figure 1 Support Vector Machine (SVM) schematic diagram

32 SVM Multiclass Classification Method Rotor vibrationfault diagnosis is a multiclass signal process problem inpractice and requires establishing a multiclass SVM classifierThe construction methods of multiclass classifier containone-against-all (1-a-a) method one-to-one (1-a-1) methodDirected a-Cyclic Graph SVM (DAGSVM) method globaloptimization classification method and so forth [13 14 18]In this paper the 1-a-a method was used to design SVMmulticlassifier A set of training samples is assumed as

119879 = (119909119894 119910119894) (119909

119897 119910119897) isin (119883 times 119884)

119897 (11)

where 119909119894isin 119883 isin R119899 and 119910

119894isin 119884 = (1 119872) 119894 = 1 2 119897

A discrimination function119891(119909) is searched inR119899 to makeeach input value 119909 have one corresponding output value 119910 Infact the essence of multiclass classification is how to find outone reasonable rule to divide all points inR119899 into119872 portionsThe SVM analytical procedure of multiclass classificationproblem based on the 1-a-1 method is drawn as follows

(1) Take the 119895th class as the positive class and the rest119872minus 1 classes as the negative class according to SVMtheory the decision functions of the 119895th class aredetermined by

119891119895(119909) = sgn (119892119895 (119909)) (12)

where 119892119895(119909) = sum119897119894=1119910119894119886119895

119894119870(119909 119909

119895) + 119887119895 119895 = 1 119872

(2) Judge the input 119909 belonging to the 119895th class where 119895is the maximum label in 1198921(119909) 119892119872(119909)

In light of the above steps a multiclass classifier ofsamples based on SVM can be built until the surplus trainingsamples are correctly classified

4 Rotor Simulation Experiment

41 Select Typical Faults Rotor imbalance shaft misalign-ment rubbing and pedestal looseness are four typical rotorvibration faults Rotor imbalance is often induced by unrea-sonable design manufacturing and fixing error attrition andso forth and involves rotormass imbalance rotor initial bend


andunbalanced coupling primarily Shaftmisalignment com-prises the misalignment of coupling and the misalignmentof bearings Contact rubbing constantly occurs due to thereduction of dynamic-static gap imbalance misalignmentand hot bend Pedestal looseness is often caused by bad fixingand long-term vibration In this paper we select the fourtypical faults to study the rotor vibration fault diagnosis basedon PPSE-SVMmethod

42 Simulation Experiment

421 Test Rig To obtain fault data rotor test rig as shownin Figure 2 is used to simulate four typical faults The rotorsimulation system includes test bench and measurementsystem On the test bench double rotor is opted and linkedby a flexible coupling Each rotor has one disk with equallydistributed holes Four acceleration sensors are installed onthe locations A B C and D of pedestal as shown in Figure 2One speed sensor is applied to measure rotor speed Thedouble rotor is driven by a motor On point D there isa bolt to simulate the rub-impact fault of rotor The mea-surement system consists of signal acquisition instrumentsignal amplifier speed controller velocity indicator andcomputer Signal acquisition instrument is used to collectthe vibration signals of acceleration and speed sensors Inaddition due to the poor quality of sensor and measuringinstrument as well as the influence of imbalance inher-ently resonance and other signals maybe existent in rotorvibration signals hained from rotor experiment Howeverthe condition hardly influence the analytical results in thispaper because the propose method (PPSE-SVM method)is to be more effective in pure fault diagnosis of rotorvibration if the method holds acceptable diagnostic precisionin the coupling fault diagnosis of rotor vibration meanwhilethe case seems to more reasonably simulate the real rotorsystem

422 Experimental Process To study the process charac-teristics of rotor vibration four kinds of typical faults(rotor imbalance shaftmisalignment pedestal looseness andrubbing) are simulated from 0 rpm to 3000 rpm on rotorvibration simulation test bench Each fault is simulated bymultiple accelerated experiments and fault data is extractedby the interval of 100 rpm sampling speed In measurementsystem four sensors (four vibration signal channels) are fixedon rotor test rig tomeasure four-point vibration accelerationsand these signals are gathered as original data of rotorvibration fault diagnosis In this process mass block is addedin the holes of disk to simulate rotor imbalance fault the shaftaxes of two rotors (S1 and S2) are not located on the sameline to simulate the shaft misalignment of rotor system Thelooseness of one (B1) or two bolts (B1 and B2) on pedestal isused to simulate the pedestal looseness fault of rotor systemThe condition of bolt contacted rotor shaft is regarded toimitate rotor rubbing fault

423 Original Data By the simulation experiment a massof original vibration signals for each typical fault is gathered

Figure 2 Rotor vibration simulation test bench and its measure-ment system

when rotor speed is from 0 to 3000 rpm with 100 rpmsampling interval One group of vibration signals has 30groups of vibration waveform under one measuring pointand different speeds Therefore in one process of speed-upor speed-down experiment 120 groups of vibration signalwaveforms for each failure modes may be collected whichreflect the process characteristics of rotor vibration faults Forpoint B three-dimensional (3-D) frequency spectrographs ofnormal state and four faults are shown in Figures 3 4 5 6and 7

43 Data Analysis From Figures 3sim7 the main feature ofrotor fault distributes in the rotate speed range (1000 rpm3000 rpm) with 100 rpm sampling interval We select thespeed band to study the proposed method (PPSE-SVMmethod) One group of vibration signals has 21 groups ofvibration waveform under one measuring point and differentspeeds Therefore in one process of speed-up or speed-down experiment 84 groups of vibration signal waveformsfor each failure modes may be collected which reflect theprocess characteristics of rotor vibration faults The speed-up (or -down) process of rotor is constituted of a numberof states Vibration shapes are different under disparaterotational speeds In fact the vibration waveforms recordthe full information of rotor vibration states under differentspeeds and time The vibration fault feature possesses somedispersiveness and randomness at each point however thevibration fault feature in rotor vibration process is regularInformation entropy matrix integrates information entropyvalues under multispeed and multichannel which reflectsthe process regularity of vibration signals Therefore theinformation entropy matrix may be employed to describethe process regularity of rotor vibration signal The originaldata which reflects the process characteristics of each faultcan be gained on rotor vibration simulation experimentAccording to PPSE method the PPSE values of vibrationwaveform signals may be calculated For one vibration faultPPSE values which fully describe the process features ofthis vibration fault under multichannel and multispeed arepromising to be obtained for constructing one PPSE matrixwhich is regarded as the fault diagnosis samples of SVMmodel


100

50

0

1 2 3 4 5 6 7 8Frequency multiplication

30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)

Figure 3 3D frequency spectrograph of rotor normal state

40020000


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)

Figure 4 3D frequency spectrograph of rotor imbalance

5 Example Analysis

51 Establish PPSE-SVMModel Thebasic idea of rotor vibra-tion fault diagnosis based on PPSE-SVM method is drawnas follows (1) gather the rotor vibration data of four typicalfaults based on rotor vibration fault simulation experiment(2) extract the vibration data characteristics (PPSE values) offour faults based on PPSE method as the training and testingsamples of SVM fault diagnosis model (3) establish SVMfault diagnosismodel by the training samples and accomplishthe rotor vibration fault diagnosis by distinguishing fault cat-egory discriminating failure severity judging fault locationand validating the robustness of PPSE-SVM method Theabove process is named fusion fault diagnosis based on PPSE-SVM method The detailed analytical procedure is shown inFigure 8

52 PPSE Feature Extraction and Parameter Selection ThePPSE values of vibration fault which reflect the process ofspeed-up or speed-down are gained based on PPSE methodandMatlab simulation software From four measuring pointsin the rotor simulation experiments four groups of PPSEsare integrated to present the variation of rotor vibrationcondition in speed-up process Each group of PPSE values ofeach channel is made up of twenty-one information entropyvalues which reflect the process variation of vibration signalat the corresponding measuring point during speed-up AllPPSE values of each vibration fault may constitute one PPSEmatrix (or PSE matrix) which is indicated by using PPSEvalues rotate speed and measuring point (channel) ThePPSE matrixes of four vibration faults are shown in Figures9 10 11 and 12

Obviously the PPSE values of each measuring pointconsist of sequential 21 entropy values Hence the 21 PPSE

60

30

0


30002500

20001500

1000500

0

Rotate

speed

(rpm

)

Am

plitu

deva

lue (

120583m

)

Figure 5 3D frequency spectrograph of shaft misalignment

200

100

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)Figure 6 3D frequency spectrograph of pedestal looseness

40

20

0


3000

2500

2000

1500

1000

500

0 Rotate

spee

d (rp

m)

Am

plitu

deva

lue (

120583m

)

Figure 7 3D frequency spectrograph of rubbing

values were lined up as a feature vector [1198641 1198642 119864

21]

which was regarded as the fault diagnosis sample of SVMmodel In light of this way 40 fault vectors fully reflectingrotor vibration process features were taken as fault samplesfor each fault mode The total number of samples of rotorvibration fault diagnosis is 160

SVM fault diagnosis model is structured by Matlaband Lib-SVM toolbox The kernel function of SVM isRadial Basis Function (RBF) kernel function (119896(119909

119894 119909) =

exp[minus119909 minus 11990911989421205902]) The optimal values of 120590 and 119862 are 007

and 136 respectively based on random search approach [3]Experiences show that the classification effect of RBF issuperior to others

53 Rotor Vibration Fault Category Diagnosis The 10 faultsamples of each vibration faultmodewere selected as trainingsamples to train SVM model and then obtain the optimalclassifications function and establish SVM diagnosis modelSelected training samples of each fault type are shown inTable 2


Rotor vibration fault simulation

Gather rotor vibration fault signals

Preprocess vibration fault signal

Extract information features (PPSE values) of eachvibration fault signal by PPSE method

Construct the PPSE vector set of rotor vibrationprocess for each vibration fault

Establish SVM fault diagnosis model

Fault diagnosis by multiclassification SVM

Output results

Trainingsamples

Classifier

Patternrecognition

Featureextraction

Testingexamples

Figure 8 Fault diagnosis procedure for rotor vibration based on PPSE-SVMmethod

Table 2 Partial PPSE vectors of each fault mode

Fault types Sample number PPSE vector [1198641 1198642 119864

21] Category number

Rotor imbalance1 4315 4373 4466 4991 4864 4443 5258 5244 5131 5101 4576 1

4513 4313 4939 4375 4919 4772 4429 4870 4982 5158

2 4398 4730 4472 4621 4713 3957 5035 5078 4997 4438 4533 13731 4770 4539 3988 4802 4325 4553 5260 4080 3659

Shaft misalignment1 5042 4966 4890 4553 5140 5145 5107 5031 5401 5431 5484 2

5020 5126 5028 5191 4949 4817 4749 4503 4586 3981

2 4790 4843 4750 4120 5327 5081 5288 5093 5338 5401 4944 25305 5488 5174 5245 4980 4939 4631 4398 3945 5217

Rotor rubbing1 5292 4867 4993 5185 5344 5081 5267 5463 5227 5469 5199 3

5119 5384 4840 5205 5618 4785 5165 5042 5368 5152

2 4910 5163 5138 4357 5230 5453 5396 5022 5360 5202 5456 35523 5570 5399 4720 3790 5061 5217 5053 5346 5137

Pedestal looseness1 3578 5260 4777 4871 4786 5275 5071 5272 5414 5142 4903 4

5315 5363 5311 5189 5019 5415 4358 5295 4175 5142

2 4505 4908 5169 5366 5029 5361 5107 5286 5567 5038 5015 45316 5233 5339 5337 5502 5332 5290 5121 5529 5270

To validate the learning ability and generalization abilityof SVM model the training samples of four faults wereinput into the SVM diagnosis model The result shows thatthese training samples are able to be completely correctlyclassified and the testing precision is 100 And then basedon the trained SVM model the remnant 30 samples of eachvibration fault mode are classified and the results are shownin Table 3

From Table 3 the mean diagnostic precision of fourtesting samples is 9667 which demonstrates that thetrained SVM model has good learning ability and good

generalization ability for rotor vibration fault classificationdiagnosis

54 Fault Degree Diagnosis In order to verify the validity ofPPSE-SVMmethod in diagnosing rotor vibration fault sever-ity rotor pedestal looseness was selected as study object Twofailure statuses of rotor pedestal looseness (resp one pedestallooseness (ie B1 looseness) and two pedestal looseness (ieB1 and B2 looseness)) were simulated to get fault data basedon rotor test rigThe PPSE values of these data were extractedand 40 groups of data of each failure status are selected


Table 3 Diagnosis result of rotor vibration fault categories

Fault types Testing sample number Correct identification number Precision Mean precisionRotor imbalance 30 29 9667

9667Shaft misalignment 30 29 9667Pedestal looseness 30 28 9333Rotor rubbing 30 30 100

PSE

valu

es

10

5

0

Rotor speeds (krmin)

3

2

1

0

Measuring points0

12

34

Figure 9 PPSE matrix of rotor imbalance fault

PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34

Figure 10 PPSE matrix of shaft misalignment fault

as fault samples respectively Similarly 10 groups of faultvectors are regarded as training samples and the surplus 30groups are looked at as testing samplesThrough training andtesting SVM diagnosis model by using the training samplesthe results show that the testing accuracy is also 100which demonstrates that the trained SVMmodel holds goodlearning ability in rotor vibration fault severity diagnosisThe remaining 30 groups of testing samples of differentfault degree are diagnosed by the trained SVM model Thediagnosis result of SVMmodel is shown in Table 4The resultillustrates that 3 samples are only mistakenly decided and thediagnosis precision is 095 Therefore PPSE-SVM method isable to availably judge rotor vibration fault degree

55 Fault Point Diagnosis Due to different measuring pointscorresponding to different locations on rotor vibration simu-lation test bench themeasuring points of sensors are selectedas study object A B C and D are assumed to represent fourmeasuring points respectively as shown in Figure 2The dataof measuring points are extracted as location fault data In

PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34

Figure 11 PPSE matrix of rotor rubbing fault

PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points

Figure 12 PPSE matrix of pedestal looseness fault

a similar way the PPSE values of these data were calculatedbased on PPSE-SVM method and 40 groups of fault data ofeach measuring point were taken as samples vectors where10 fault data were taken as training samples and the rest of 30fault data were looked at as test samples for each measuringpoint Through rotor vibration fault point diagnosis theresults show that the testing accuracy is 100 and the averagediagnosis precision is 9333 which is shown in Table 5 Itis indicated that the PPSE-SVM diagnosis method is feasibleand effective in rotor vibration fault point diagnosis

56 Robustness Verification To validate the robustness ofPPSE-SVMmethod in rotor vibration fault diagnoses includ-ing judgments of fault category severity and location theoriginal signals of the aforementioned test samples of threefault types were overlapped by Gaussian white noise withmean value 0 and variance 5 And then the PPSE valuesof these original signals of vibration fault were gained asnew test samples based on PPSE method Ultimately thesenew test samples of three fault types were inputted in


Table 4 Diagnosis results of fault severity

Fault degree Test sample number Correct identification number Precision Mean precisionOne pedestal looseness 30 29 9667 95Two pedestal looseness 30 28 9333

Table 5 Diagnosis results of rotor vibration fault locations

Fault points Test sample number Correct identification number Precision Mean precisionA 30 28 9333

9333B 30 29 9667C 30 28 9333D 30 27 90

Table 6 Verification results of the robustness of PPSE-SVMmethod in rotor vibration fault category severity and locations

Fault type Test sample number Correct number Precision Reduction precision Mean precisionCategory 120 115 9583 083

9433Severity 60 56 9333 167Location 120 112 9333 0

the corresponding SVM fault diagnosis models trained inSections 53sim55 The analysis results are shown in Table 6

As shown in Table 6 after overlapping Gaussian noiseon the original vibration signals of test samples the diag-nosis precisions of three fault types (category severity andlocation) are 09583 09333 and 09333 respectively andonly reduce by 00083 00167 and 0 relative to those beforesuperposition severally so that the mean precision of rotorvibration fault diagnosis comes to 09433 The results revealthat the proposed PPSE-SVMmethod has good fault-tolerantcapability and strong robustness in antinoise interference

6 Conclusion

Theobjective of these efforts is attempted to advance a processfault diagnosis methodmdashProcess Power Spectrum Entropyand Support Vector Machine (PPSE-SVM) methodmdashby fus-ing the advantages of information entropy method and SVMtheory for rotor vibration fault diagnosis from a processperspective based on information fusion techniqueThroughrotor vibration fault diagnosis based on PPSE-SVMmethodsome conclusions are drawn as follows

(1) The Process Power Spectrum Entropy (PPSE) valuescan effectively reflect the process variation of rotorvibration signals

(2) PPSE-SVM model can be established by small sam-ples PPSE feature vectors extracted from the faultvibration data of rotor fault vibration simulationexperiments

(3) The PPSE-SVM model trained by PPSE feature vec-tors is demonstrated to be an efficient fault diagnosismodel which possesses strong learning ability gen-eralization ability and fault tolerance ability becauseof high testing precision (100) high diagnosis pre-cision (resp 09667 095 and 09333) and strong

antinoise interference ability (09433 in precision)in the rotor vibration fault diagnosis and analysison fault category failure severity fault points androbustness from a process perspective

(4) The presented PPSE-SVM method is also proved tobe effective and reasonable and this study providesa promising diagnosis technology for rotor vibrationfault

(5) Some idealized factors were considered for rotorvibration fault diagnosis based on the PPSE-SVMmethod on rotor vibration simulation test bench inthis paper For complex machinery like an aero-engine the validity of the presented PPSE-SVMmethod needs to be further verified in vibration faultdiagnosis

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper is supported by the National Natural ScienceFoundation of China (Grant nos 51175017 and 51275024) andResearch Foundation for the Doctoral Program of HigherEducation (no 2011112110011) The authors would like tothank them

References

[1] D J Ewins ldquoControl of vibration and resonance in aero enginesand rotating machinerymdashan overviewrdquo International Journal ofPressure Vessels and Piping vol 87 no 9 pp 504ndash510 2010


[2] F Al-Badour M Sunar and L Cheded ldquoVibration analysis ofrotating machinery using time-frequency analysis and wavelettechniquesrdquo Mechanical Systems and Signal Processing vol 25no 6 pp 2083ndash2101 2011

[3] C W Fei and G C Bai ldquoWavelet correlation feature scaleentropy and fuzzy support vector machine approach for aero-engine whole-body vibration fault diagnosisrdquo Shock and Vibra-tion vol 20 no 2 pp 341ndash349 2013

[4] P W Tse S Gontarz and X J Wang ldquoEnhanced eigenvectoralgorithm for recovering multiple sources of vibration signalsin machine fault diagnosisrdquo Mechanical Systems and SignalProcessing vol 21 no 7 pp 2794ndash2813 2007

[5] J-D Wu and C-H Liu ldquoAn expert system for fault diagnosisin internal combustion engines using wavelet packet transformand neural networkrdquo Expert Systems with Applications vol 36no 3 pp 4278ndash4286 2009

[6] L A Overbey and M D Todd ldquoDynamic system changedetection using a modification of the transfer entropyrdquo Journalof Sound and Vibration vol 322 no 1-2 pp 438ndash453 2009

[7] L Qu L Li and J Lee ldquoEnhanced diagnostic certainty usinginformation entropy theoryrdquoAdvanced Engineering Informaticsvol 17 no 3-4 pp 141ndash150 2003

[8] H Cui L Zhang R Kang and X Lan ldquoResearch on faultdiagnosis for reciprocating compressor valve using informationentropy and SVM methodrdquo Journal of Loss Prevention in theProcess Industries vol 22 no 6 pp 864ndash867 2009

[9] X Xing ldquoPhysical entropy information entropy and theirevolution equationsrdquo Science in China A Mathematics PhysicsAstronomy vol 44 no 10 pp 1331ndash1339 2001

[10] H Endo and R B Randall ldquoEnhancement of autoregressivemodel based gear tooth fault detection technique by the useof minimum entropy deconvolution filterrdquo Mechanical Systemsand Signal Processing vol 21 no 2 pp 906ndash919 2007

[11] JWang Y T Ai X F Liu et al ldquoStudy on quantitative diagnosismethod of rotor vibration faults based on process informationentropyrdquo in Proceedings of the 2nd International Conference onComputer Engineering and Technology (ICCET rsquo10) pp V5-407ndashV5-411 Chengdu China April 2010

[12] J Ye ldquoFault diagnosis of turbine based on fuzzy cross entropy ofvague setsrdquo Expert Systems with Applications vol 36 no 4 pp8103ndash8106 2009

[13] N Garcıa-Pedrajas and D Ortiz-Boyer ldquoAn empirical study ofbinary classifier fusion methods for multiclass classificationrdquoInformation Fusion vol 12 no 2 pp 111ndash130 2011

[14] W Ding and J Yuan ldquoSpike sorting based on multi-classsupport vectormachine with superposition resolutionrdquoMedicaland Biological Engineering and Computing vol 46 no 2 pp139ndash145 2008

[15] Z-W Guo and G-C Bai ldquoClassification using least squaressupport vector machine for reliability analysisrdquo Applied Mathe-matics and Mechanics vol 30 no 7 pp 853ndash864 2009

[16] AWidodo andB-S Yang ldquoSupport vectormachine inmachinecondition monitoring and fault diagnosisrdquo Mechanical Systemsand Signal Processing vol 21 no 6 pp 2560ndash2574 2007

[17] S-F Yuan and F-L Chu ldquoSupport vector machines-based faultdiagnosis for turbo-pump rotorrdquoMechanical Systems and SignalProcessing vol 20 no 4 pp 939ndash952 2006

[18] M Saimurugan K I Ramachandran V Sugumaran and NR Sakthivel ldquoMulti component fault diagnosis of rotationalmechanical system based on decision tree and support vectormachinerdquo Expert Systems with Applications vol 38 no 4 pp3819ndash3826 2011

[19] W T Peter Y H Peng and R Yam ldquoWavelet analysis andenvelop detection for rolling element bearing fault diagnosistheir effectives and flexibilitiesrdquo Journal of Vibration and Acous-tics vol 12 no 3 pp 303ndash310 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



risk minimization instead of empirical risk minimization tosolve small-sample nonlinear high-dimensional problemsand so on SVMhas been proved to embody various strengthsof complete theory good adaptability global optimizationshort training time and good generalization ability in faultdiagnosis [3 8 14ndash18]

The purpose of this present study attempts to proposean effective approach which is Process Power SpectrumEntropy-SVM (PPSE-SVM) method for rotor vibration faultdiagnosis based on PPSE method and SVM theory Thismethod is promising to resolve the randomization of vibra-tion fault and the difficulty of extracting vibration faultsamples Based on rotor vibration simulation workbenchfour typical faults are simulated and their vibration dataare gained The PPSE values (ie information features) ofthese data are calculated based on PPSE method to constructeigenvectors as fault diagnosis samples SVM fault diagnosismodel was established based on the fault diagnosis samplesThe feasibility and validity of PPSE-SVMmethod are verifiedby diagnosing rotor vibration fault types failure severity faultpoint and noise immunity

2 Process Power Spectrum Entropy Method

Information entropy was first propounded by Shannon toevaluate information capacity [3 6ndash12] Assuming thatM is aLebesgue space with an algebra 120575 generated by a measurableset 119867 and a measure 120583 (120583(M) = 1) and the space Mmay be decomposed as a limited partitioning A = (119860

119894)

which is an incompatible set satisfying M = ⋃119899

119894=1119860119894and

119860119894⋂119860119895= 0 forall119894 = 119895 based on information entropy theory the

information entropy E(A) of A is denoted as [3]

119864 (A) = minus119899

sum

119894=1

120583 (119860119894) log120583 (119860

119894) (1)

where 120583(119860119894) is the measurement of sample 119860

119894 119894 = 1 2 119899

As shown in (1) information entropy 119864 shows actuallythe chaotic degree of uncertain factors in a system Moredisorder and randomness of the system always lead to that thecorresponding information entropy values become greaterand vice versa [7]

The Power Spectrum Entropy (PSE) values of rotor vibra-tion signal can be extracted from the features of vibrationsignals in frequency domain from an energy perspectivewhen 119883(120596) is the discrete Fourier transform of a single-channel signal 119909

119905 as follows

119883 (120596) =1

2119899120587

119873minus1

sum

119905=1

119909119905119890minus119895120596119905

(2)

Thus the power spectrum of119883(120596) is defined by

119878 (120596) =1

2120587119873|119883 (120596)|

2 (3)

due to the energy conserve law in the signal transformationprocess from the time domain to the frequency domain [11]that is

sum1199092(119905) Δ119905 = sum |119883 (120596)|

2Δ120596 (4)

Equation (4) reveals that the total energy of signal equalsthe sum of the subenergy of each frequency componentTherefore the power entropy 119878 = 119878

1 1198782 119878

119873 of every

natural frequency is regarded as one original signal partitionthe corresponding information entropy (also called as PPSE)can be defined by

119864119891= minus

119873

sum

119894=1

119902119894log 119902119894 (5)

where the subscript 119891 stands for the frequency domain 119902119894is

the ratio of the 119894th power spectrum to the whole spectrumwhich is denoted by

119902119894=

119878119894

sum119873

119894=1119878119894

(6)

Equation (5) is called the Power Spectrum Entropy (PSE) ofsignal 119878 When PSE is applied to fault diagnosis the methodis called PSE method PSE explains the spectrum structureof single-channel vibration signal The more uniformity thevibration energy distributes in the whole frequency compo-sition the more complex the signal is and the greater theuncertainty degree is When the process of rotor speed-up(or -down) is constituted from multiple rotation speeds thePSE values of different rotation speeds describe the vibrationcondition of whole speed-up (or -down) process Hence PSEmethod is called also Process PSE (PPSE) method

Due to the lack of quantitative indexes for informa-tion entropy information entropy values cannot be directlyapplied to diagnose rotor fault The PPSE presenting theconditions of rotor vibration is promising to be used todistinguish rotor faults However the different momentsand measuring points of vibration waveform always resultin that PPSE values have a certain distribution range foreach fault and there is invariably some overlap amongthe PPSE distribution ranges of different faults [7ndash9] Forexample the PPSE values of four typical faults from multiplesimulation experiments are listed and shown in Table 1Theirdistribution ranges have large overlap regions When thePPSE value of an unknown fault is 50 it is difficult to judge towhich class the unknown fault belongs Therefore a perfectand effective identification method needs to be proposedto address this issue SVM method is an intelligent patterrecognition method and is promising to address this overlapproblem of the PPSE values of different fault modes

3 SVM Method

31 Fundamental Theory Support Vector Machine (SVM)[15ndash17] is a newmachine learningmethodwhich is promisingto address the realistic issues of small sample nonlinear andhigh-dimensional pattern recognition in fault diagnosis andhas the advantages of complete theory good adaptabilityglobal optimization short training time and good general-ization ability SVM adopts small support vectors in trainingsamples standing for the whole vector set to establish a SVMclassifier for fault diagnosis Assuming that the fault trainingsample set is (119909

119894 119910119894) 119894 = 1 2 119871 where 119909 isin 119877

119899 and





min 1

2(120596 sdot 120596) + 119862

119899

sum

119894=1

120577119894

st 119910119894[(120596 sdot 119909

119894) + 119887] + 120577

119894ge 1

(7)





hyperplane is


119886119894119910119894119909119894sdot 119909 + 119887

lowast) (8)


119891 (119909) = sgn(sum119894

119886lowast

119894119910119894119896 (119909119894 119909) + 119887

lowast) (9)



119896 (119909119894 119909) = exp[minus

1003817100381710038171003817119909 minus 11990911989410038171003817100381710038172

1205902] (10)


y

y1a1lowast

y2a2lowast y3a3

lowast ysaslowast





Φ

y = sgn(s

sumi=1




119879 = (119909119894 119910119894) (119909

119897 119910119897) isin (119883 times 119884)

119897 (11)


119894isin 119884 = (1 119872) 119894 = 1 2 119897



119891119895(119909) = sgn (119892119895 (119909)) (12)

where 119892119895(119909) = sum119897119894=1119910119894119886119895

119894119870(119909 119909

119895) + 119887119895 119895 = 1 119872















100

50

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


40020000


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


5 Example Analysis




60

30

0


30002500

20001500

1000500

0

Rotate

speed

(rpm

)

Am

plitu

deva

lue (

120583m

)


200

100

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m


40

20

0


3000

2500

2000

1500

1000

500

0 Rotate

spee

d (rp

m)

Am

plitu

deva

lue (

120583m

)



21]



119894 119909) =












Output results

Trainingsamples

Classifier

Patternrecognition

Featureextraction

Testingexamples




21] Category number

Rotor imbalance1 4315 4373 4466 4991 4864 4443 5258 5244 5131 5101 4576 1

4513 4313 4939 4375 4919 4772 4429 4870 4982 5158

2 4398 4730 4472 4621 4713 3957 5035 5078 4997 4438 4533 13731 4770 4539 3988 4802 4325 4553 5260 4080 3659


5020 5126 5028 5191 4949 4817 4749 4503 4586 3981

2 4790 4843 4750 4120 5327 5081 5288 5093 5338 5401 4944 25305 5488 5174 5245 4980 4939 4631 4398 3945 5217

Rotor rubbing1 5292 4867 4993 5185 5344 5081 5267 5463 5227 5469 5199 3

5119 5384 4840 5205 5618 4785 5165 5042 5368 5152

2 4910 5163 5138 4357 5230 5453 5396 5022 5360 5202 5456 35523 5570 5399 4720 3790 5061 5217 5053 5346 5137


5315 5363 5311 5189 5019 5415 4358 5295 4175 5142

2 4505 4908 5169 5366 5029 5361 5107 5286 5567 5038 5015 45316 5233 5339 5337 5502 5332 5290 5121 5529 5270









PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34




PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points









9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













min 1

2(120596 sdot 120596) + 119862

119899

sum

119894=1

120577119894

st 119910119894[(120596 sdot 119909

119894) + 119887] + 120577

119894ge 1

(7)





hyperplane is


119886119894119910119894119909119894sdot 119909 + 119887

lowast) (8)


119891 (119909) = sgn(sum119894

119886lowast

119894119910119894119896 (119909119894 119909) + 119887

lowast) (9)



119896 (119909119894 119909) = exp[minus

1003817100381710038171003817119909 minus 11990911989410038171003817100381710038172

1205902] (10)


y

y1a1lowast

y2a2lowast y3a3

lowast ysaslowast





Φ

y = sgn(s

sumi=1




119879 = (119909119894 119910119894) (119909

119897 119910119897) isin (119883 times 119884)

119897 (11)


119894isin 119884 = (1 119872) 119894 = 1 2 119897



119891119895(119909) = sgn (119892119895 (119909)) (12)

where 119892119895(119909) = sum119897119894=1119910119894119886119895

119894119870(119909 119909

119895) + 119887119895 119895 = 1 119872















100

50

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


40020000


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


5 Example Analysis




60

30

0


30002500

20001500

1000500

0

Rotate

speed

(rpm

)

Am

plitu

deva

lue (

120583m

)


200

100

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m


40

20

0


3000

2500

2000

1500

1000

500

0 Rotate

spee

d (rp

m)

Am

plitu

deva

lue (

120583m

)



21]



119894 119909) =












Output results

Trainingsamples

Classifier

Patternrecognition

Featureextraction

Testingexamples




21] Category number

Rotor imbalance1 4315 4373 4466 4991 4864 4443 5258 5244 5131 5101 4576 1

4513 4313 4939 4375 4919 4772 4429 4870 4982 5158

2 4398 4730 4472 4621 4713 3957 5035 5078 4997 4438 4533 13731 4770 4539 3988 4802 4325 4553 5260 4080 3659


5020 5126 5028 5191 4949 4817 4749 4503 4586 3981

2 4790 4843 4750 4120 5327 5081 5288 5093 5338 5401 4944 25305 5488 5174 5245 4980 4939 4631 4398 3945 5217

Rotor rubbing1 5292 4867 4993 5185 5344 5081 5267 5463 5227 5469 5199 3

5119 5384 4840 5205 5618 4785 5165 5042 5368 5152

2 4910 5163 5138 4357 5230 5453 5396 5022 5360 5202 5456 35523 5570 5399 4720 3790 5061 5217 5053 5346 5137


5315 5363 5311 5189 5019 5415 4358 5295 4175 5142

2 4505 4908 5169 5366 5029 5361 5107 5286 5567 5038 5015 45316 5233 5339 5337 5502 5332 5290 5121 5529 5270









PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34




PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points









9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















100

50

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


40020000


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m

)


5 Example Analysis




60

30

0


30002500

20001500

1000500

0

Rotate

speed

(rpm

)

Am

plitu

deva

lue (

120583m

)


200

100

0


30002500

20001500

1000500

0

Rotate

speed (r

pm)

Am

plitu

deva

lue (

120583m


40

20

0


3000

2500

2000

1500

1000

500

0 Rotate

spee

d (rp

m)

Am

plitu

deva

lue (

120583m

)



21]



119894 119909) =












Output results

Trainingsamples

Classifier

Patternrecognition

Featureextraction

Testingexamples




21] Category number

Rotor imbalance1 4315 4373 4466 4991 4864 4443 5258 5244 5131 5101 4576 1

4513 4313 4939 4375 4919 4772 4429 4870 4982 5158

2 4398 4730 4472 4621 4713 3957 5035 5078 4997 4438 4533 13731 4770 4539 3988 4802 4325 4553 5260 4080 3659


5020 5126 5028 5191 4949 4817 4749 4503 4586 3981

2 4790 4843 4750 4120 5327 5081 5288 5093 5338 5401 4944 25305 5488 5174 5245 4980 4939 4631 4398 3945 5217

Rotor rubbing1 5292 4867 4993 5185 5344 5081 5267 5463 5227 5469 5199 3

5119 5384 4840 5205 5618 4785 5165 5042 5368 5152

2 4910 5163 5138 4357 5230 5453 5396 5022 5360 5202 5456 35523 5570 5399 4720 3790 5061 5217 5053 5346 5137


5315 5363 5311 5189 5019 5415 4358 5295 4175 5142

2 4505 4908 5169 5366 5029 5361 5107 5286 5567 5038 5015 45316 5233 5339 5337 5502 5332 5290 5121 5529 5270









PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34




PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points









9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















Output results

Trainingsamples

Classifier

Patternrecognition

Featureextraction

Testingexamples




21] Category number

Rotor imbalance1 4315 4373 4466 4991 4864 4443 5258 5244 5131 5101 4576 1

4513 4313 4939 4375 4919 4772 4429 4870 4982 5158

2 4398 4730 4472 4621 4713 3957 5035 5078 4997 4438 4533 13731 4770 4539 3988 4802 4325 4553 5260 4080 3659


5020 5126 5028 5191 4949 4817 4749 4503 4586 3981

2 4790 4843 4750 4120 5327 5081 5288 5093 5338 5401 4944 25305 5488 5174 5245 4980 4939 4631 4398 3945 5217

Rotor rubbing1 5292 4867 4993 5185 5344 5081 5267 5463 5227 5469 5199 3

5119 5384 4840 5205 5618 4785 5165 5042 5368 5152

2 4910 5163 5138 4357 5230 5453 5396 5022 5360 5202 5456 35523 5570 5399 4720 3790 5061 5217 5053 5346 5137


5315 5363 5311 5189 5019 5415 4358 5295 4175 5142

2 4505 4908 5169 5366 5029 5361 5107 5286 5567 5038 5015 45316 5233 5339 5337 5502 5332 5290 5121 5529 5270









PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34




PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points









9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

Rotor speed (krmin)

30

20

10

0

Measuring points0

12

34




PSE

valu

es

10

5

0


3

2

1

0

Measuring points0

12

34


PSE

valu

es

10

5

0

10

5

0


3

2

1

0Measuring points









9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














9333B 30 29 9667C 30 28 9333D 30 27 90






6 Conclusion










Acknowledgments


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article quantitative diagnosis of rotor vibration...

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